Part II
Circuit-Level Parallelism
Part II
Circuit-Level
Parallelism
Winter 2014
Sorting and
Searching
Numerical
Computations
Parallel Processing, Circuit-Level Parallelism
7. Sorting and Selection Networks
8A. Search Acceleration Circuits
6B. Arithmetic and Counting Circuits
6C. Fourier Transform Circuits
Slide 1
About This Presentation
This presentation is intended to support the use of the textbook
Introduction to Parallel Processing: Algorithms and Architectures
(Plenum Press, 1999, ISBN 0-306-45970-1). It was prepared by
the author in connection with teaching the graduate-level course
ECE 254B: Advanced Computer Architecture: Parallel Processing,
at the University of California, Santa Barbara. Instructors can use
these slides in classroom teaching and for other educational
purposes. Any other use is strictly prohibited. © Behrooz Parhami
Edition
First
Winter 2014
Released
Revised
Revised
Revised
Spring 2005
Spring 2006
Fall 2008
Fall 2010
Winter 2013
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 2
II Circuit-Level Parallelism
Circuit-level specs: most realistic parallel computation model
• Concrete circuit model; incorporates hardware details
• Allows realistic speed and cost comparisons
• Useful for stand-alone systems or acceleration units
Topics in This Part
Chapter 7 Sorting and Selection Networks
Chapter 8A Search Acceleration Circuits
Chapter 8B Arithmetic and Counting Circuits
Chapter 8C Fourier Transform Circuits
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 3
7 Sorting and Selection Networks
Become familiar with the circuit model of parallel processing:
• Go from algorithm to architecture, not vice versa
• Use a familiar problem to study various trade-offs
Topics in This Chapter
7.1 What is a Sorting Network?
7.2 Figures of Merit for Sorting Networks
7.3 Design of Sorting Networks
7.4 Batcher Sorting Networks
7.5 Other Classes of Sorting Networks
7.6 Selection Networks
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 4
7.1 What is a Sorting Network?
x0
x1
x2
.
.
.
n-sort er
.
.
.
x n–1
y0
y1
y2
The outputs are a
permutation of the
inputs satisfying
y0 Š y1 Š ... Šy n–1
(non-descending)
y n–1
input 0
min
Fig. 7.1 An n-input
sorting network or
an n-sorter.
in
out
in
out
in
out
in
out
2-sorter
input 1
max
Block Diagram
Alternate Representations
Fig. 7.2 Block diagram and four different
schematic representations for a 2-sorter.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 5
Building Blocks for Sorting Networks
input 0
Implementation with
bit-parallel inputs
min
in
2-sorter
2-sorter
out
in with
Implementation
bit-serial inputs
in
Block Diagram
b
out
0
k
1
min(a, b )
b <a?
k
0
k
1
max(a, b )
Alternate Representations
a
0
MSB-first serial inputs
k
Compare
in
max
input 1
a
out
out
min(a, b )
1
S
R
Q
S
R
Q
b <a?
a<b ?
0
b
1
max(a, b )
Reset
Fig. 7.3 Parallel and bit-serial hardware realizations of a 2-sorter.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 6
Proving a Sorting Network Correct
x0
y0
3
2
1
1
x1
y1
2
3
3
2
x2
y2
5
1
2
3
x3
y3
1
5
5
5
Fig. 7.4
Block diagram and schematic representation of a 4-sorter.
Method 1: Exhaustive test – Try all n! possible input orders
Method 2: Ad hoc proof – for the example above, note that y0 is smallest,
y3 is largest, and the last comparator sorts the other two outputs
Method 3: Use the zero-one principle – A comparison-based sorting
algorithm is correct iff it correctly sorts all 0-1 sequences (2n tests)
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 7
Elaboration on the Zero-One Principle
0
1
1
0
1
0
3
6
9
1
8
5
Invalid
6-sort er
1
3
6*
5*
8
9
0
0
1
0
1
1
Deriving a 0-1 sequence that is not correctly sorted, given an
arbitrary sequence that is not correctly sorted.
Let outputs yi and yi+1 be out of order, that is yi > yi+1
Replace inputs that are strictly less than yi with 0s and all others with 1s
The resulting 0-1 sequence will not be correctly sorted either
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 8
7.2 Figures of Merit for Sorting Networks
Cost: Number of
comparators
In the following example, we have 5 comparators
Delay: Number
of levels
The following 4-sorter has 3 comparator levels on
its critical path
Cost  Delay
The cost-delay product for this example is 15
x0
y0
3
2
1
1
x1
y1
2
3
3
2
x2
y2
5
1
2
3
x3
y3
1
5
5
5
Fig. 7.4
Winter 2014
Block diagram and schematic representation of a 4-sorter.
Parallel Processing, Circuit-Level Parallelism
Slide 9
Cost as a Figure of Merit
Optimal size is known for n = 1 to 8:
0, 1, 3, 5, 9, 12, 16, 19
n = 6, 12 modules, 5 levels
n = 9, 25 modules , 9 l evel s n = 9, 25 modules, 9 levels
n = 10, 29 modules, 9 level s
n = 1 2, 3 9 mo d u les, 9 levels
Fig. 7.5 Some low-cost
sorting networks.
Winter 2014
n = 16, 60 modules, 10 l evels
Parallel Processing, Circuit-Level Parallelism
Slide 10
Delay as a Figure of Merit
Optimal delay is known for n = 1 to 10:
0, 1, 3, 3, 5, 5, 6, 6, 7, 7
n = 6, 12 m odules, 5 levels
These 3 comparators
constitute one level
n = 9 , 25 mo d u les, 8 l ev els
n = 10, 31 modules , 7 l evel s
n = 12, 40 modules , 8 levels
Fig. 7.6 Some fast
sorting networks.
Winter 2014
n = 16, 61 modul es , 9 l evels
Parallel Processing, Circuit-Level Parallelism
Slide 11
Cost-Delay Product as a Figure of Merit
n = 6, 12 m odules, 5 levels
n = 9 , 25 mo d u les, 8 l ev els
n = 10, 29 modules, 9 level s
Low-cost 10-sorter from Fig. 7.5
Cost  Delay = 29  9 = 261
n = 10, 31 modules , 7 l evel s
Fast 10-sorter from Fig. 7.6
Cost  Delay = 31  7 = 217
The most cost-effective n-sorter may be neither
the fastest design, nor the lowest-cost design
n = 12, 40 modules , 8 levels
n = 16, 61 modul es , 9 l evels
n = 16, 60 modules, 10 l evels
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 12
7.3 Design of Sorting Networks
C(n)
D(n )
Cost  Delay
=
=
=
n(n – 1)/2
n
n2(n – 1)/2 = Q(n3)
Rotate
Rotate
by
bydegrees
90
90
todegrees
see the
odd-even
exchange
patterns
Fig. 7.7 Brick-wall 6-sorter based on odd–even transposition.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 13
Insertion Sort and Selection Sort
x0
x1
x2
.
.
.
x n–2
x n–1
(n–1)- sorte r
Inse rtion sort
y0
y1
y2
.
.
.
x0
x1
x2
.
.
.
y n–2
y n–1
.
.
.
x n–2
x n–1
y0
y1
y2
(n–1)- sorte r
.
.
.
y n–2
y n–1
Sele ction sor t
P aral lel i nserti on sort = P aral lel s el ecti on s ort = Parall el bubble s ort!
C(n) = n(n – 1)/2
D(n ) = 2n – 3
Cost  Delay
= Q(n3)
Fig. 7.8 Sorting network based on insertion sort or selection sort.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 14
Theoretically Optimal Sorting Networks
O(log n) depth
x0
x1
x2
.
.
.
O(n log n)
n-sort er
size
.
.
.
x n–1
y0
y1
y2
Note that even for these
optimal networks, delay-cost
product is suboptimal; but
this
the best we
canado
Theis outputs
are
permutation of the
Existing
networks
inputssorting
satisfying
2 n) latency and
have
y0 Š O(log
y1 Š ...
Šy n–1
2
O(n
log n) cost
(non-descending)
y n–1 Given that log2 n is only 20
AKS sorting network
(Ajtai, Komlos, Szemeredi: 1983)
for n = 1 000 000, the latter
are more practical
Unfortunately, AKS networks are not practical owing to large (4-digit)
constant factors involved; improvements since 1983 not enough
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 15
7.4 Batcher Sorting Networks
x0
First
sorted
sequence x
x1
x2
w0
v1
x3
y0
w1
v2
y1
Second
sorted
sequence y
(2, 3)-merger
v0
y2
w2
w3
v4
y5
y6
Fig. 7.9
Winter 2014
(1, 1)
(1, 2)
v3
y3
y4
a0
a1
b0
b1
b2
w4
v5
(2, 4)-merger
(2, 3)-merger
(1, 2)-merger
c0
d0
d1
(1, 1)
Batcher’s even–odd merging network for 4 + 7 inputs.
Parallel Processing, Circuit-Level Parallelism
Slide 16
Proof of Batcher’s Even-Odd Merge
x0
v0
x1
First
sorted
sequence x
x2
v1
x3
w1
y0
x has k 0s
y has k  0s
w2
y2
v3
y3
w3
v has keven = k/2 + k /2 0s
w4
w has kodd = k/2 + k /2 0s
v4
y4
y5
y6
Assume:
v2
y1
Second
sorted
sequence y
Use the zero-one principle
w0
v5
(2, 4)-merger
(2, 3)-merger
Case a: keven = kodd
v
w
0
Case b: keven = kodd+1
v
w
0
Case c: keven = kodd+2
v
w
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
1
1
1
1
0
0
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
Parallel Processing, Circuit-Level Parallelism
1
1
1
1
Out of order
Winter 2014
1
Slide 17
1
1
Batcher’s Even-Odd Merge Sorting
Batcher’s (m, m) even-odd merger,
for m a power of 2:
.
.
.
n/2-sorter
.
.
.
.
.
.
D(m) = D(m/2) + 1 = log2 m + 1
(n/2, n/2)merger
.
.
.
n/2-sorter
.
.
.
C(m) = 2C(m/2) + m – 1
= (m – 1) + 2(m/2 – 1) + 4(m/4 – 1) + . . .
= m log2m + 1
Cost  Delay = Q(m log2 m)
.
.
.
Fig. 7.10 The recursive
structure of Batcher’s even–
odd merge sorting network.
Batcher sorting networks based on the
even-odd merge technique:
C(n) = 2C(n/2) + (n/2)(log2(n/2)) + 1
 n(log2n)2/ 2
D(n) = D(n/2) + log2(n/2) + 1
= D(n/2) + log2n
= log2n (log2n + 1)/2
Cost  Delay = Q(n log4n)
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 18
Example Batcher’s Even-Odd 8-Sorter
.
.
.
n/2-sorter
.
.
.
.
.
.
(n/2, n/2)merger
.
.
.
n/2-sorter
.
.
.
.
.
.
4 -s orters
Even
(2 ,2 )-merger
Od d
(2 ,2 )-merger
Fig. 7.11 Batcher’s even-odd merge
sorting network for eight inputs .
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 19
Bitonic-Sequence Sorter
Bitonic sequence:
Shifted
right half
1 3 3 4 6 6 6 2 2 1 0 0
Rises, then falls
0
1
2
. . .
Shift right half of
data to left half
(superimpose the
two halves)
Bitonic
sequence
n/2
. . .
n–1
In eac h position,
keep the smaller
value of each pair
and ship the larger
value to the right
8 7 7 6 6 6 5 4 6 8 8 9
Falls, then rises
Each half is a bitonic
sequence that can be
sorted independently
8 9 8 7 7 6 6 6 5 4 6 8
The previous sequence,
right-rotated by 2
Winter 2014
0
1
2
. . .
n/2
. . .
n–1
Fig. 14.2 Sorting a bitonic
sequence on a linear array.
Parallel Processing, Circuit-Level Parallelism
Slide 20
Batcher’s Bitonic Sorting Networks
2-input 4-input bit onicsorters sequence sorters
Fig. 7.12 The recursive
structure of Batcher’s
bitonic sorting network.
Winter 2014
8-input bit onicsequence sorter
Fig. 7.13 Batcher’s
bitonic sorting network
for eight inputs.
Parallel Processing, Circuit-Level Parallelism
Slide 21
7.5 Other Classes of Sorting Networks
Fig. 7.14 Periodic balanced
sorting network for eight inputs.
Winter 2014
Desirable properties:
a. Regular / modular
(easier VLSI layout).
b. Simpler circuits via
reusing the blocks
c. With an extra block
tolerates some faults
(missed exchanges)
d. With 2 extra blocks
provides tolerance to
single faults (a missed
or incorrect exchange)
e. Multiple passes
through faulty network
(graceful degradation)
f. Single-block design
becomes fault-tolerant
by using an extra stage
Parallel Processing, Circuit-Level Parallelism
Slide 22
Shearsort-Based Sorting Networks (1)
0
1
2
3
4
5
6
7
0
1
2
3
7
6 5 4
Corresponding
2-D mesh
Snake-like
row sorts
Fig. 7.15
Winter 2014
Column
sorts
Snake-like
row sorts
Design of an 8-sorter based on shearsort on 24 mesh.
Parallel Processing, Circuit-Level Parallelism
Slide 23
Shearsort-Based Sorting Networks (2)
0
1
2
3
4
5
6
7
0
1
3
2
4
5
7
6
Corresponding
2-D mesh
Right
Left
column column
sort
sort
Snake-like row sort
Fig. 7.16
Winter 2014
Right
Left
column column
sort
sort
Snake-like row sort
Some of the same
advantages as
periodic balanced
sorting networks
Design of an 8-sorter based on shearsort on 24 mesh.
Parallel Processing, Circuit-Level Parallelism
Slide 24
7.6 Selection Networks
Direct design may
yield simpler/faster
selection networks
3rd smallest element
Can remove
this block if
smallest three
inputs needed
4-s orters
Even
(2,2)-merger
Odd
(2,2)-merger
Can remove
these four
comparators
Deriving an (8, 3)-selector from Batcher’s even-odd merge 8-sorter.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 25
Categories of Selection Networks
Unfortunately we know even less about selection networks
than we do about sorting networks.
One can define three selection problems [Knut81]:
I. Select the k smallest values; present in sorted order
II. Select kth smallest value
III. Select the k smallest values; present in any order
Circuit and time complexity: (I) hardest, (III) easiest
Type-I
8 inputs
Winter 2014
(8, 4)-selector
Smallest
2nd smallest
3rd smallest
4th smallest
Type-II
4th smallest
Parallel Processing, Circuit-Level Parallelism
Type-III
The 4
smallest
In any
order
Slide 26
Type-III Selection Networks
Figure 7.17
Winter 2014
A type III (8, 4)-selector.
8-Classifier
Parallel Processing, Circuit-Level Parallelism
Slide 27
Classifier Networks
Classifiers:
Selectors that separate
the smaller half of values
from the larger half
8 inputs
Smaller
4 values
8-Classifier
Larger
4 values
Use of classifiers for building sorting networks
2-Classifier
4-Classifier
2-Classifier
8-Classifier
2-Classifier
4-Classifier
2-Classifier
Problem: Given O(log n)-time and O(n log n)-cost n-classifier designs,
what are the delay and cost of the resulting sorting network?
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 28
8A Search Acceleration Circuits
Much of sorting is done to facilitate/accelerate searching
• Simple search can be speeded up via special circuits
• More complicated searches: range, approximate-match
Topics in This Chapter
8A.1 Systolic Priority Queues
8A.2 Searching and Dictionary Operations
8A.3 Tree-Structured Dictionary Machines
8A.4 Associative Memories
8A.5 Associative Processors
8A.6 VLSI Trade-offs in Search Processors
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 29
8A.1 Systolic Priority Queues
Problem: We want to maintain a large list of keys, so that we can
add new keys into it (insert operation) and obtain the smallest key
(extract operation) whenever desired.
Unsorted list:
Constant-time insertion / Linear-time extraction
Sorted list:
Linear-time insertion / Constant-time extraction
Can both insert and extract operations (priority-queue operations)
be performed in constant time, independent of the size of the list?
Priority queue
5 2 8 6
1
3 7 9 1 4
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 30
5 2 8 6 3 7 9 1 4
5 2 8 6 3 7 9 1 4
5 2 8 6 3 7 9 1 4
5 2 8 6 3 7 1 9
5 2 8 6 3 1 7
5 2 8 6 1 3
5 2 8 1 6
5 2 1 8
5 1 2
Insertion of new keys and
5
1
read-out of the smallest
First Attempt:
Via a LinearArray Sorter
key value can be done in
constant time, but the
“hole” created by the
extracted value cannot be
filled in constant time
Fig. 2.9
Winter 2014
4
4
4
3
3
3
2
9
7
4
6
8
3
5
9
7
4
4
4
3
9
7
6
8
4
9
7
6
7
8
9
2
1
2
3
1
2
3
4
1
2
3
4
5
1
2
3
4
5
6
1
2
3
4
5
6
4
Parallel Processing, Circuit-Level Parallelism
6
5
7
9
1
5
6
9
7
6
8
7
6
9
8
7
9
8
7
9
8
7
9
8
9
Slide 31
5 -- 2 -- 8 -- 6 -- 3 -- 7 -- 9 -- 1 -- 4
5 -- 2 -- 8 -- 6 -- 3 -- 7 -- 9 -- 1 -- 4
5 -- 2 -- 8 -- 6 -- 3 -- 7 -- 9 -- 1 4
5 -- 2 -- 8 -- 6 -- 3 -- 7 -- 9 -- 1 4
5 -- 2 -- 8 -- 6 -- 3 -- 7 -- 9 1
4
5 -- 2 -- 8 -- 6 -- 3 -- 7 -- 1 9 4
A Viable
Systolic
Priority
Queue
9
5 -- 2 -- 8 -- 6 -- 3 -- 7 1
4
5 -- 2 -- 8 -- 6 -- 3 -- 1 7 4
9
7
5 -- 2 -- 8 -- 6 -- 3 1
4
9
9
Extract
1
4
7
4
Operating on
every other
clock cycle,
allows holes
to be filled
Winter 2014
Extract
4
7
7
7
Extract
7
9
9
9
9
5 -- 2 -- 8 -- 6 -- 3 9
5 -- 2 -- 8 -- 6 -- 3 9
Parallel Processing, Circuit-Level Parallelism
Slide 32
Systolic Data Structures
Fig. 8.3
Each node holds the smallest (S),
median (M),and largest (L)
value in its subtree
Each subtree is balanced
or has one fewer element on the
left (root flag shows this)
S L
M
S L
M
S L
M
S L
M
S L
M
S L
M
S L
M
5
[5, 87]
19 or 20
items
Winter 2014
176
87
S L
M
Example: 20 elements,
3 in root, 8 on left,
and 9 on right
S L
M
S L
M
[87, 176]
20 items
Systolic data structure
for minimum, maximum,
and median finding.
S L
M
Insert
Insert
Insert
Insert
S L
M
S L
M
2
20
127
195
S L
M
Extractmin
Extractmed
Extractmax
Parallel Processing, Circuit-Level Parallelism
8 elements:
3+2+3
S L
M
Update/access
examples for the
systolic data
structure of Fig. 8.3
Slide 33
8A.2 Searching and Dictionary Operations
Parallel (p + 1)-ary search on PRAM
logp+1(n + 1)
= log2(n + 1) / log2(p + 1)
= Q(log n / log p) steps
0
1
2
Speedup  log p
Optimal: no comparison-based
Example:
search algorithm can be faster
8
Example:
n = 26, p = 2
P0
P0
P1
P1
P0
n = 26
A single search in a sorted list
can’t be significantly speeded
p = 2 up
through parallel processing,
but all hope is not lost:
17
P1
25
Step Step Step
2
1
0
Dynamic data (sorting overhead)
Batch searching (multiple lookups)
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 34
Dictionary Operations
Basic dictionary operations: record keys x0, x1, . . . , xn–1
search(y)
insert(y, z)
delete(y)
Find record with key y; return its associated data
Augment list with a record: key = y, data = z
Remove record with key y; return its associated data
Some or all of the following operations might also be of interest:
findmin
findmax
findmed
findbest(y)
findnext(y)
findprev(y)
extractmin
extractmax
extractmed
Find record with smallest key; return data
Find record with largest key; return data
Find record with median key; return data
Find record with key “nearest” to y
Find record whose key is right after y in sorted order
Find record whose key is right before y in sorted order
Remove record(s) with min key; return data
Remove record(s) with max key; return data
Remove record(s) with median key value; return data
Priority queue operations: findmin, extractmin (or findmax, extractmax)
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 35
8A.3 Tree-Structured Dictionary Machines
Search 2
Search 1
x0
Pipelined
search
Input Root
"Ci rcle"
Tree
x1
x2
search(y): Pass OR
of match signals &
data from “yes” side
x3
x4
x5
x6
"Tri angle"
Tree
Output Root
Fig. 8.1
Winter 2014
Combining in the
triangular nodes
A tree-structured dictionary machine.
Parallel Processing, Circuit-Level Parallelism
x7
findmin / findmax:
Pass smaller / larger
of two keys & data
findmed:
Not supported here
findbest(y): Pass
the larger of two
match-degree
indicators along with
associated record
Slide 36
Insertion and Deletion in the Tree Machine
In p ut Ro ot
in s ert(y ,z)
Counters keep track
of the vacancies
in each subtree
1
0
0
*
0
1
0
1
*
0
Deletion needs second
pass to update the
vacancy counters
2
0
0
*
*
0
2
1
1
*
Redundant
insertion (update?)
and deletion (no-op?)
Implementation:
Merge the circle and
triangle trees by folding
Ou tp ut Roo t
Figure 8.2
Winter 2014
Tree machine storing 5 records and containing 3 free slots.
Parallel Processing, Circuit-Level Parallelism
Slide 37
Physical Realization of a Tree Machine
Tree machine in folded form
Inner node
Leaf node
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 38
VLSI Layout of a Tree
H-tree layout (used, e.g., for clock distribution network in high-performance microchips)
A clock
domain
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 39
8A.4 Associative Memories
Associative or content-addressable memories (AMs, CAMs)
Binary (BCAM) vs. ternary (TCAM)
d
d
Image source: http://www.pagiamtzis.com/cam/camintro.html
Mismatch in cell connects the match line (ml) to ground
If all cells in the word match the input pattern, a word match is indicated
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 40
Word Match Circuitry
The match line is precharged and then pulled down by any mismatch
Image source: http://www.pagiamtzis.com/cam/camintro.html
Note that each CAM cell is nearly twice as complex as an SRAM cell
More transistors, more wires
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 41
CAM Array Operation
Image source: http://www.pagiamtzis.com/cam/camintro.html
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 42
Current CAM Applications
Packet forwarding
Routing tables specify the path to be taken by matching an incoming
destination address with stored address prefixes
Prefixes must be stored in order of decreasing length (difficult updating)
Packet classification
Determine packet category based on information in multiple fields
Different classes of packets may be treated differently
Associative caches / TLBs
Main processor caches are usually not fully associative (too large)
Smaller specialized caches and TLBs benefit from full associativity
Data compression
Frequently used substrings are identified and replaced by short codes
Substring matching is accelerated by CAM
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 43
History of Associative Processing
Associative memory
Parallel masked search of all words
Bit-serial implementation with RAM
100111010110001101000
Associative processor
Add more processing logic to PEs
Table 4.1
Comparand
Mask
Memory
array with
comparison
logic
Entering the second half-century of associative processing
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Decade Events and Advances
Technology
Performance
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
1940s
Formulation of need & concept
Relays
1950s
Emergence of cell technologies
Magnetic, Cryogenic
Mega-bit-OPS
1960s
Introduction of basic architectures Transistors
1970s
Commercialization & applications
ICs
Giga-bit-OPS
1980s
Focus on system/software issues
VLSI
Tera-bit-OPS
1990s
Scalable & flexible architectures
ULSI, WSI
Peta-bit-OPS
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 44
8A.5 Associative Processors
Associative or
content-addressable
memories/processors
constituted early
forms of SIMD
parallel processing
Fig. 23.1
Functional view of
an associative
memory/processor.
Winter 2014
Control
Unit
Global Operat ions Cont rol & Response
Read
Lines
Comparand
Mask
Data and
Commands
Broadcast
Response
Store
(T ags)
Cell 0
t0
Cell 1
t1
Cell 2
t2
.
.
.
.
.
.
Cell m–1
Parallel Processing, Circuit-Level Parallelism
Global T ag
Operations
Unit
.
.
.
t m –1
Slide 45
Search Functions in Associative Devices
Exact match: Locating data based on partial knowledge of contents
Inexact match: Finding numerically or logically proximate values
Membership: Identifying all members of a specified set
Relational: Determining values that are less than, less than or equal, etc.
Interval: Marking items that are between or outside given limits
Extrema: Finding the maximum, minimum, next higher, or next lower
Rank-based: Selecting kth or k largest/smallest elements
Ordered retrieval: Repeated max- or min-finding with elimination (sorting)
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 46
Classification of Associative Devices
Handling of bits
within words
Parallel
Serial
Parallel
WPBP:
Fully
parallel
WPBS:
Bitserial
Serial
WSBP:
Wordserial
WSBS:
Fully
serial
Handling
of words
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 47
WSBP: Word-Serial Associative Devices
Strictly speaking, this is not a parallel processor, but with superhigh-speed
shift registers and deeply pipelined processing logic, it behaves like one
From processing logic
Superhigh-speed shift registers
Processing
logic
One word
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 48
WPBS: Bit-Serial Associative Devices
One bit of every word is processed in one device cycle
Advantages:
1. Can be implemented with conventional memory
2. Easy to add other capabilities beyond search
PE
PE
Example: Adding field A to field B in every
word, storing the sum in field S
PE
Memory
array
PE
PE
PE
PE
PE
One bit-slice
Winter 2014
Loop:
Read next bit slice of A
Read next bit slice of B
(carry from previous slice is in PE flag C)
Find sum bits; store in next bit slice of S
Find new carries; store in PE flag
Endloop
Parallel Processing, Circuit-Level Parallelism
Slide 49
Goodyear STARAN Associative Processor
First computer
based on
associative
memory (1972)
Aimed at air
traffic control
applications
Aircraft conflict
detection is an
O(n2) operation
AM can do it in
O(n) time
Winter 2014
256 PEs
Parallel Processing, Circuit-Level Parallelism
Slide 50
Flip Network Permutations in the Goodyear STARAN
The 256 bits in a bit-slice could be routed to 256
PEs in different arrangements (permutations)
Figs. in this slide from J. Potter, “The STARAN
Architecture and Its Applications …,” 1978 NCC
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 51
Distributed Array Processor (DAP)
To neig h bo ri ng
p ro cess o rs
N
E
W
S
Condition
A
{
NN
Fro m
n ei gh b orin g E E
p ro cess o rs S S
WW
Fro m
con trol
u ni t
Fig. 23.6
The bit-serial
processor of
DAP.
Winter 2014
{
Row
Col
C
Mu x
Q
To ro w/col
resp o ns es
Ful l Car ry
add er Sum
Mu x
Memo ry
D
S
Fro m so u th n ei gh b or
Parallel Processing, Circuit-Level Parallelism
To no rt h neig h bo r
Slide 52
DAP’s High-Level Structure
N
W
Column j
E
S
Program
memory
Row i
Master
control
unit
Processors
Q plane
C plane
Fast I/O
Register Q in
processor ij
A plane
D plane
Host
interface
unit
One plane
of memory
Host
work station
Array memory
(at least
32K planes)
Winter 2014
Local memory
for processor ij
Fig. 23.7 The high-level
architecture of DAP system.
Parallel Processing, Circuit-Level Parallelism
Slide 53
8A.6 VLSI Trade-offs in Search Processors
This section has not been written yet
References:
[Parh90] B. Parhami, "Massively Parallel Search Processors: History and
Modern Trends," Proc. 4th Int'l Parallel Processing Symp., pp. 91-104, 1990.
[Parh91] B. Parhami, "Scalable Architectures for VLSI-Based Associative
Memories," in Parallel Architectures, ed. by N. Rishe, S. Navathe, and D. Tal,
IEEE Computer Society Press, 1991, pp. 181-200.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 54
8B Arithmetic and Counting Circuits
Many parallel processing techniques originate from, or find
applications in, designing high-speed arithmetic circuits
• Counting, addition/subtraction, multiplication, division
• Limits on performance and various VLSI trade-offs
Topics in This Chapter
8B.1 Basic Addition and Counting
8B.2 Circuits for Parallel Counting
8B.3 Addition as a Prefix Computation
8B.4 Parallel Prefix Networks
8B.5 Multiplication and Squaring Circuits
8B.6 Division and Square-Rooting Circuits
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 55
8B.1 Basic Addition and Counting
y
Fig. 5.3 (in Computer Arithmetic)
Using full-adders in building
bit-serial and ripple-carry adders.
x
xi
Shift
Carry
FF
yi
ci
ci+1
FA
Clock
Shift
si
s
(a) Bit-serial adder.
x31
y31
c32
x1
c31
FA
. . .
y1
c2
x0
FA
c0
FA
cout
s32
Ideal cost:
O(k)
y0
c1
cin
s31
s1
Ideal latency:
O(log k)
Can these be
achieved
simultaneously?
s0
(b) Ripple-carry adder.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 56
Constant-Time Counters
Any fast adder design can be specialized and optimized to yield
a fast counter (carry-lookahead, carry-skip, etc.)
One can use redundant representation to build a constant-time
counter, but a conversion penalty must be paid during read-out
Count regis ter divided i nto three stages
Load
1
Load
Increment
1
Incr em e nter
Incr em e nter
Control
2
Control
1
Fig. 5.12 (in Computer Arithmetic)
Fast (constant-time) three-stage up counter.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Counting is
fundamentally
simpler than
addition
Slide 57
8B.2 Circuits for Parallel Counting
1-bit full-adder = (3; 2)-counter
FA
FA
1
1
0
1
FA
1 0
0
FA
Circuit reducing 7 bits to their
3-bit sum = (7; 3)-counter
2
FA
1
0
0
HA
1
2
3-bit
ripple-carry
adder
FA
HA
3
Circuit reducing n bits to their
log2(n + 1)-bit sum
= (n; log2(n + 1))-counter
Winter 2014
2
1
0
Fig. 8.16 (in Computer Arithmetic)
A 10-input parallel counter also
known as a (10; 4)-counter.
Parallel Processing, Circuit-Level Parallelism
Slide 58
Accumulative Parallel Counters
n increment signals vi, 2q–1 < n  2q
True generalization of
sequential counters
FA
q-bit initial
count x
FA
n
increment
signals vi
q-bit final count y = x + Svi
Possible application:
Compare Hamming weight
of a vector to a constant
Winter 2014
FA
Count
register
FA
Parallel
incrementer
FA
FA
q-bit tally of up to 2q – 1
of the increment signals
FA
FA
FA
FA
FA
FA
FA
FA
q-bit
initial
count x
cq
Latency:
Ignore, or use
O(log
n)
for decision
Cost:
O(n)
FA
q-bit final count y
(q + 1)-bit final count y
Parallel Processing, Circuit-Level Parallelism
Slide 59
8B.3 Addition as a Prefix Computation
Example: Prefix sums
x0
x1
x2
x0
x0 + x1
x0 + x1 + x2
s0
s1
s2
. . .
. . .
. . .
xi
x0 + x1 + . . . + xi
si
Sequential time with one processor is O(n)
Simple pipelining does not help

xi
Function unit
Fig. 8.4
Winter 2014
si
Latches
si
xi
Four-stage pipeline
Prefix computation using a latched or pipelined function unit.
Parallel Processing, Circuit-Level Parallelism
Slide 60
Improving the Performance with Pipelining
Ignoring pipelining overhead, it appears that we
have achieved a speedup of 4 with 3 “processors.”
Can you explain this anomaly? (Problem 8.6a)
xi–6³ a[i–7]
xi–7
a[i–6]
xi–1
a[i–1]
s i–12
x[i – 12]
Delays
Del ays
Delay
Del ay
xi
a[i]
Function uni t
comput ing
 xi–10 ³ a[i–11]
xi–11
 xi–9
a[i–8] x³ i–8
a[i–9]
³ a[i–10]
 xi–5
a[i–4] ³xi–4
a[i–5]
Fig. 8.5 High-throughput prefix computation using a pipelined function unit.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 61
Carry Determination as a Prefix Computation
gi pi
Carry is:
0
0
1
1
annihilated or killed
propagated
generated
(impossible)
0
1
0
1
g k1 p k1
gk1 pk1
ck
ck
xi
g k2 p k2
c k1
gi = xi yi
pi = xi  yi
g i+1 p i+1
gi
pi
...
...
gk2 pk2
g1
...
ck1
yi
ck2
c2
p1
g0
gk2 pk2
c
Carry network
c
c
c 1 ck
...
c k2
p0 pk1
gk1
ci
c i+1
Figure from Computer Arithmetic
g1
p1
...
0
k1
g1 p1
k2
c2
c1
...
c1
g0 p0
g0
g–1=
p–1=0
c0
p0
c0
c0
si
Fig. 5.15 (ripple-carry network) superimposed on Fig. 5.14 (generic adder).
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 62
8B.4 Parallel Prefix Networks
x n–1 x n–2
...
x3 x2 x1 x0
+
+
+
T(n) = T(n/2) + 2
= 2 log2n – 1
C(n) = C(n/2) + n – 1
= 2n – 2 – log2n
Prefix Sum n/2
+
s n–1 s n–2
+
...
s3 s2 s1 s0
This is the Brent-Kung
Parallel prefix network
(its delay is actually
2 log2n – 2)
Fig. 8.6 Prefix sum network built of one n/2-input network and n – 1 adders.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 63
Example of Brent-Kung Parallel Prefix Network
x15 x14 x13 x12 x
11
x10 x9 x8 x x x x x x x x
7
6 5
4 3
2 1
0
Originally developed
by Brent and Kung as
part of a VLSI-friendly
carry lookahead adder
One level
of latency
T(n) = 2 log2n – 2
s 15 s 14 s 13 s 12 s
Fig. 8.8
Winter 2014
11
s 10 s 9 s 8 s s s s s s s s
7
6 5
4 3
2 1
0
C(n) = 2n – 2 – log2n
Brent–Kung parallel prefix graph for n = 16.
Parallel Processing, Circuit-Level Parallelism
Slide 64
Another Divide-and-Conquer Design
Ladner-Fischer construction
xn–1
xn/2
P refix Sum n/2
s n–1
+
s n/2
T(n) = T(n/2) + 1
= log2n
C(n) = 2C(n/2) + n/2
P refix Sum n/2
= (n/2) log2n
...
...
...
x0
...
...
+
xn/2–1
s n/2–1
s0
Simple Ladner-Fisher
Parallel prefix network
(its delay is optimal,
but has fan-out issues
if implemented directly)
Fig. 8.7 Prefix sum network built of two n/2-input networks and n/2 adders.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 65
Example of Kogge-Stone Parallel Prefix Network
x15 x14 x13 x12 x x x x x x x x x x x x
11 10 9
8 7
6 5
4 3
2 1
0
T(n) = log2n
C(n) = (n – 1) + (n – 2)
+ (n – 4) + . . . + n/2
= n log2n – n – 1
Optimal in delay,
but too complex
in number of cells
and wiring pattern
s15 s 14 s 13 s 12 s s s s s s s s s s s s
11 10 9
8 7
6 5
4 3
2 1
0
Fig. 8.9
Winter 2014
Kogge-Stone parallel prefix graph for n = 16.
Parallel Processing, Circuit-Level Parallelism
Slide 66
Comparison and Hybrid Parallel Prefix Networks
x15 x14 x13 x12 x x x x x x x x x x x x
11 10 9
8
7
6
5
4
3
2
1
0
x15 x14 x13 x12 x x x x x x x x x x x x
11 10 9
8
7
6
5
4
3
2
1
0
Brent/Kung
6 levels
26 cells
s 15 s 14 s 13 s 12 s s s s s s s s s s s s
11 10 9
8
7
6
5
4
3
2
1
0
Kogge/Stone
4 levels
49 cells
s15 s 14 s 13 s 12 s s s s s s s s s s s s
11 10 9
8
7
6
5
4
3
2
1
0
x15 x14 x13 x12 x x x x x x x x x x x x
11 10 9
8 7
6 5
4 3
2 1
0
BrentKung
Fig. 8.10 A hybrid
Brent–Kung /
Kogge–Stone
parallel prefix
graph for n = 16.
KoggeStone
Han/Carlson
5 levels
Brent- 32 cells
Kung
s15 s14 s13 s12 s
Winter 2014
11
s10 s9 s8 s7 s6 s5 s4 s3 s2 s1 s0
Parallel Processing, Circuit-Level Parallelism
Slide 67
Linear-Cost, Optimal Ladner-Fischer Networks
Define a type-x parallel prefix network as one that:
Produces the leftmost output in optimal log2 n time
Yields all other outputs with at most x additional delay
Note that even the
Brent-Kung network
produces the leftmost
output in optimal time
Type -0
xn–1
xn/2
P refix Sum n/2
+
P refix Sum n/2 Type -1
...
...
We are interested in
building a type-0 overall
network, but can use
type-x networks (x > 0)
as component parts
x0
...
...
Type -0
xn/2–1
+
s n/2–1
s0
...
s n–1
s n/2
Recursive construction of the fastest possible
parallel prefix network (type-0)
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 68
Examples of Type-0, 1, 2 Parallel Prefix Networks
x15 x14 x13 x12 x x x x x x x x x x x x
11 10 9
8
7
6
5
4
3
2
1
0
x15 x14 x13 x12 x x x x x x x x x x x x
11 10 9
8
7
6
5
4
3
2
1
0
Brent/Kung:
16-input type-2
network
s 15 s 14 s 13 s 12 s s s s s s s s s s s s
11 10 9
8
7
6
5
4
3
2
1
0
Kogge/Stone
16-input type-0
network
s15 s 14 s 13 s 12 s s s s s s s s s s s s
11 10 9
8
7
6
5
4
3
2
1
0
x15 x14 x13 x12 x x x x x x x x x x x x
11 10 9
8 7
6 5
4 3
2 1
0
BrentKung
Fig. 8.10 A hybrid
Brent–Kung /
Kogge–Stone
parallel prefix
graph for n = 16.
KoggeStone
Han/Carlson
16-input type-1
Brent- network
Kung
s15 s14 s13 s12 s
Winter 2014
11
s10 s9 s8 s7 s6 s5 s4 s3 s2 s1 s0
Parallel Processing, Circuit-Level Parallelism
Slide 69
8B.5 Multiplication and Squaring Circuits
Notation for our discussion of multiplication algorithms:
a
x
p
Multiplicand
Multiplier
Product (a  x)
p2k–1p2k–2
ak–1ak–2 . . . a1a0
xk–1xk–2 . . . x1x0
. . . p3 p2 p1 p0
Initially, we assume unsigned operands
Sequential:
O(k) circuit
complexity
O(k) time with
carry-save
additions

a
x
Multiplicand
Multiplier
x 0 a 20
x 1 a 21
x 2 a 22
x 3 a 23
Partial
products
bit-matrix
p
Product
Parallel:
O(k2)
circuit
complexity
O(log k)
time
Fig. 9.1 (in Computer Arithmetic) Multiplication of 4-bit binary numbers.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 70
Divide-and-Conquer (Recursive) Multipliers
Building wide multiplier from narrower ones

aH
aL
xH
xL
Rearranged part ial products
in 2b-by-2b mult iplicat ion
2b bits
a L xL
a L xH
a H xL
a H xL
a H xH
a L xL
a L xH
a H xH
3b bits
p
Fig. 12.1 (in Computer Arithmetic)
Divide-and-conquer (recursive)
strategy for synthesizing a 2b  2b
multiplier from b  b multipliers.
Winter 2014
b bit s
C(k) = 4C(k/2) + O(k) = O(k2)
T(k) = T(k/2) + O(log k) = O(log2 k)
Parallel Processing, Circuit-Level Parallelism
Slide 71
Karatsuba Multiplication
2b  2b
multiplication requires four b  b multiplications:
(2baH + aL)  (2bxH + xL) = 22baHxH + 2b (aHxL + aLxH) + aLxL
Karatsuba noted that one of the four multiplications can be removed
at the expense of introducing a few additions:
(2baH + aL)  (2bxH + xL) =
b bits
22baHxH + 2b [(aH + aL)  (xH + xL) – aHxH – aLxL] + aLxL
Mult 1
Mult 3
Benefit is quite significant for
extremely wide operands
Winter 2014
aH
aL
xH
xL
Mult 2
C(k) = 3C(k/2) + O(k) = O(k1.585)
T(k) = T(k/2) + O(log k) = O(log2 k)
Parallel Processing, Circuit-Level Parallelism
Slide 72
Divide-and-Conquer Squarers
Building wide squarers from narrower ones

axH
axLL
xH
xL
Rearranged part ial products
in 2b-by-2b mult iplicat ion
2b bits
axLL x L
axL x H
axH
H xL
axHH x L
axHH x H
xaLL x L
axL x H
xaHH x H
p
b bit s
3b bits
Divide-and-conquer (recursive) strategy for synthesizing a
2b  2b squarer from b  b squarers and multiplier.
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 73
VLSI Complexity Issues and Bounds
Any VLSI circuit computing the product of two k-bit integers must
satisfy the following constraints:
AT grows at least as fast as k3/2
AT2 is at least proportional to k2
Array multipliers: O(k2) gate count and area, O(k) time
AT =
O(k3)
AT2 =
O(k4)
Simple recursive multipliers: O(k2) gate count, O(log2 k) time
AT =
O(k2 log2 k) ?
AT2 =
O(k2 log4 k) ?
Karatsuba multipliers: O(k1.585) gate count, O(log2 k) time
AT =
O(k1.585 log2 k) ?
AT2 =
O(k1.585 log4 k) ???
Discrepancy due to the fact that interconnect area is not taken into
account in our previous analyses
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 74
Theoretically Best Multipliers
Schonhage and Strassen (via FFT); best result until 2007
O(log k) time
O(k log k log log k) complexity
In 2007, M. Furer managed to replace the log log k term with an
asymptotically smaller term
It is an open problem whether there exist logarithmic-delay
multipliers with linear cost
(it is widely believed that there are not)
In the absence of a linear cost multiplication circuit, multiplication
must be viewed as a more difficult problem than addition
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 75
8B.6 Division and Square-Rooting Circuits
Division via Newton’s method: O(log k) multiplications
Using Schonhage and Strassen’s FFT-based multiplication, leads to:
O(log2 k) time
O(k log k log log k) complexity
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 76
Theoretically Best Dividers
Best known bounds; cannot be achieved at the same time (yet)
O(log k) time
O(k log k log log k) complexity
In 1966, S. A. Cook established these simultaneous bounds:
O(log2 k) time
O(k log k log log k) complexity
In 1983, J. H. Reif reduced the time complexity to the current best
O(log k (log log k)2) time
In 1984, Beame/Cook/Hoover established these simultaneous bounds:
O(log k) time
O(k4) complexity
Given our current state of knowledge, division must be viewed as
a more difficult problem than multiplication
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 77
Implications for Ultrawide High-Radix Arithmetic
Arithmetic results with k-bit binary operands hold with no change
when the k bits are processed as g radix-2h digits (gh = k)
k bits
h-bit
group
g groups
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 78
Another Circuit Model: Artificial Neural Nets
Artificial
neuron
Activation
function
Supervised learning
Feedforward network
Three layers: input, hidden, output
No feedback
Output
Inputs Weights
Threshold
Characterized by connection
topology and learning method
Recurrent network
Simple version due to Elman
Feedback from hidden nodes to special nodes at the input layer
Hopfield network
All connections are bidirectional
Winter 2014
Diagrams from
http://www.learnartificialneuralnetworks.com/
Parallel Processing, Circuit-Level Parallelism
Slide 79
8C Fourier Transform Circuits
Fourier transform is quite important, and it also serves as a
template for other types of arithmetic-intensive computations
• FFT; properties that allow efficient implementation
• General methods of mapping flow graphs to hardware
Topics in This Chapter
8C.1 The Discrete Fourier Transform
8C.2 Fast Fourier Transform (FFT)
8C.3 The Butterfly FFT Network
8C.4 Mapping of Flow Graphs to Hardware
8C.5 The Shuffle-Exchange Network
8C.6 Other Mappings of the FFT Flow Graph
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 80
8C.1 The Discrete Fourier Transform
y0
y1
y2
x0
x1
x2
.
.
.
xn–1
DFT
.
.
.
yn–1
x0
x1
x2
Inv.
DFT
.
.
.
xn–1
n–point DFT
x in time domain
y in frequency domain
Some operations are
easier in frequency
domain; hence the
need for transform
Other important transforms for discrete signals:
z-transform (generalized form of Fourier transform)
Discrete cosine transform (used in JPEG image compression)
Haar transform (a wavelet transform, which like DFT, has a fast version)
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 81
Defining the DFT and Inverse DFT
DFT yields output sequence yi based on input sequence xi (0  i < n)
yi =
∑j=0 to n–1wnij xj
O(n2)-time naïve algorithm
where wn is the nth primitive root of unity; wnn = 1, wnj ≠ 1 (1  j < n)
Examples:
w4 = i
w3 = (1 + i 3)/2
w8 = 2(1 + i )/2
The inverse DFT is almost exactly the same computation:
xi = (1/n) ∑j=0 to n–1wnij yj
Input seq. xi (0  i < n) is said to be in time domain
Output seq. yi (0  i < n) is the input’s frequency-domain characterization
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 82
DFT of a Cosine Waveform
DFT of a cosine
with a frequency
1/10 the sampling
frequency fs
Frequency fs
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 83
DFT of a Cosine with Varying Resolutions
DFT of a cosine
with a frequency
1/10 the sampling
frequency fs
Frequency fs
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 84
DFT as Vector-Matrix Multiplication
DFT and inverse DFT computable
via matrix-by-vector multiplication
yi =
n–1
∑j = 0
Y
=
W

X
wnij xj
DFT matrix
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 85
Application of DFT to Smoothing or Filtering
Input signal with noise
DFT
Low-pass filt er
Inverse DFT
Recovered smoot h signal
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 86
DFT Application Example
Signal corrupted by 0-mean
random noise
FFT shows strong frequency
components of 50 and 120
The uncorrupted signal was:
x = 0.7 sin(2p 50t) + sin(2p 120t)
Source of images:
http://www.mathworks.com/help/techdoc/ref/fft.html
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 87
Application of DFT to Spectral Analysis
697 Hz
1
2
3
A
770 Hz
4
5
6
B
852 Hz
7
8
9
C
Received tone
DFT
941 Hz
*
1209
Hz
0
#
1336 1477
Hz
Hz
D
1633
Hz
T one frequency assignment s
for t ouch-t one dialing
Frequency spect rum of received tone
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 88
8C.2 Fast Fourier Transform
DFT yields output sequence yi based on input sequence xi (0  i < n)
yi =
∑j=0 to n–1wnij xj
i
Fast Fourier Transform (FFT):
The Cooley-Tukey algorithm
j
O(n log n)-time DFT algorithm that derives y
from half-length sequences u and v that are DFTs
of even- and odd-indexed inputs, respectively
yi = ui + wni vi
(0  i < n/2)
yi+n/2
= ui + wni+n/2 vi = ui – wni vi
T(n) = 2T(n/2) + n = n log2n
T(n) = T(n/2) + 1 = log2n
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Image from Wikipedia
Butterfly
operation
sequentially
in parallel
Slide 89
More General Factoring-Based Algorithm
Image from
Wikipedia
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 90
8C.3 The Butterfly FFT Network
u: DFT of even-indexed inputs
v: DFT of odd-indexed inputs
yi = ui + wni vi
(0  i < n/2)
yi+n/2
= ui + wni+n/2 vi
x0
u0
y0
x0
u0
y0
x4
u1
y1
x1
u2
y4
x2
u2
y2
x2
u1
y2
x6
u3
y3
x3
u3
y6
x1
v0
y4
x4
v0
y1
x5
v1
y5
x5
v2
y5
x3
v2
y6
x6
v1
y3
x7
v3
y7
x7
v3
y7
Fig. 8.11
Winter 2014
Butterfly network for an 8-point FFT.
Parallel Processing, Circuit-Level Parallelism
Slide 91
Butterfly Processor
Performs a pair of multiply-add operations,
where the multiplication is by a constant
processor

Winter 2014
+
–
Design can be optimized
by merging the adder
and subtractor, as they
receive the same inputs
Parallel Processing, Circuit-Level Parallelism
Slide 92
Computation Scheme for 16-Point FFT
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
0
0
0
0
4
0
2
4
6
0
1
2
3
4
5
6
7
0
0
0
4
0
0
Bit-reversal
permutation
Winter 2014
0
4
0
4
0
2
4
6
Butterfly
operation
a
b j
Parallel Processing, Circuit-Level Parallelism
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
a + b wj
a  b wj
Slide 93
8C.4 Mapping of Flow Graphs to Hardware
Given a computation flow graph, it can be mapped to hardware
a
b
(a + b) c d
ef
+

c
d

e
f


/
z
Direct
one-to-one
mapping
(possibly
with
pipelining)
Latch positions in a four-stage pipeline
+


Time

t=0
/
Pipelining period
Fig. 25.6
Winter 2014
Output
available

Latency
Parhami’s textbook on computer arithmetic.
Parallel Processing, Circuit-Level Parallelism
Slide 94
Ad-hoc Scheduling on a Given Set of Resources
Given a computation flow graph, it can be mapped to hardware
a
b
++
c
d

e
f
–
(a + b) c d
ef

/

z
Assume:
tadd = 1
tmult = 3
tdiv = 8
tsqrt = 10
Latch positions in a four-stage pipeline
0 1 2 3
+
Add

Pipelining period
Winter 2014
Time
14

Multt = 0
Div /
Sqrt
6
24
Time

Output
available
/

Latency
Parallel Processing, Circuit-Level Parallelism
Slide 95
Mapping through Projection
Given a flow graph, it can be projected in various directions to obtain
corresponding hardware realizations
Multiple nodes of a flow graph may map onto a single hardware node
That one hardware node then performs the computations associated
with the flow graph nodes one by one, according to some timing
arrangement (schedule)
x0
u0
y0
x0
u0
y0
x4
u1
y1
x1
u2
y4
x2
u2
y2
x2
u1
y2
x6
u3
y3
x3
u3
y6
x1
v0
y4
x4
v0
y1
x5
v1
y5
x5
v2
y5
x3
v2
y6
x6
v1
y3
x7
v3
y7
x7
v3
y7
Winter 2014
Projection direction
Parallel Processing, Circuit-Level Parallelism
Linear array,
with each cell
acting for one
butterfly
network row
Slide 96
8C.5 The Shuffle-Exchange Network
x0
u0
y0
x0
u0
y0
x4
u1
y1
x1
u2
y4
x2
u2
y2
x2
u1
y2
x6
u3
y3
x3
u3
y6
x1
v0
y4
x4
v0
y1
x5
v1
y5
x5
v2
y5
x3
v2
y6
x6
v1
y3
x7
v3
y7
x7
v3
y7
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 97
Variants of the Butterfly Architecture
x0
u0
y0
x1
u2
y4
x2
u1
y2
x3
u3
y6
x4
v0
y1
x5
v2
y5
x6
v1
y3
x7
v3
y7
Fig. 8.12
Winter 2014
FFT network variant and its shared-hardware realization.
Parallel Processing, Circuit-Level Parallelism
Slide 98
8C.6 Other Mappings of the FFT Flow Graph
This section is incomplete at this time
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 99
More
Economical
FFT
Hardware
x0
u0
y0
x1
u2
y4
x2
u1
y2
x3
u3
y6
x4
v0
y1
x5
v2
y5
x6
v1
y3
x7
v3
y7
Project
Project
Project
Fig. 8.13
Linear array of
log2n cells for
n-point FFT
computation.
Winter 2014
x
a

Project
1
a+b
y
0
win
b

ab
Control
0

1
Parallel Processing, Circuit-Level Parallelism
Slide 100
Space-Time Diagram for the Feedback FFT Array
Feedback
butterfly
processor
Winter 2014
Parallel Processing, Circuit-Level Parallelism
Slide 101